[-- Attachment #1.1: Type: text/plain, Size: 514 bytes --] Can we prove that the map A -> loop susp A is (k+1)-connected for a k-connected A, without having to invoke Freudenthal? -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/81b245ba-e70f-4a13-8d0c-4eaad69f3da8%40googlegroups.com. [-- Attachment #1.2: Type: text/html, Size: 945 bytes --] <div dir="ltr">Can we prove that the map A -> loop susp A is (k+1)-connected for a k-connected A, without having to invoke Freudenthal?<span style="left: 119.999px; top: 753.619px; font-size: 15.7742px; font-family: sans-serif; transform: scaleX(0.876455);"></span></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/81b245ba-e70f-4a13-8d0c-4eaad69f3da8%40googlegroups.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/81b245ba-e70f-4a13-8d0c-4eaad69f3da8%40googlegroups.com</a>.<br />

[-- Attachment #1.1: Type: text/plain, Size: 1292 bytes --] Here is an observation I had made: We have that A is k-connected, we can show that susp A is (k+1)-connected and then show that loop susp A is k-connected. This gives us a k-connected map A -> 1 and a k-connected map loop susp A -> 1. This gives us a diagram which commutes with A and loop susp A in the top corners and 1 in the bottom. The LHS composition is homotopic to the RHS composition hence naming eta : A -> loop susp A, we have eta o unitmap being k-connected hence eta must also be k-connected. This isn't quite there. Now I was hoping to use the fact that loop spaces are pullbacks hence there are maps coming out of 1s hence (k+1)-connectedness appears, but I couldn't get it to work. On Sunday, 4 August 2019 12:59:09 UTC+3, Ali Caglayan wrote: > > Can we prove that the map A -> loop susp A is (k+1)-connected for a > k-connected A, without having to invoke Freudenthal? > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/a4549eab-f563-42ad-a95a-10166e8e0664%40googlegroups.com. [-- Attachment #1.2: Type: text/html, Size: 1862 bytes --] <div dir="ltr">Here is an observation I had made:<div><br></div><div>We have that A is k-connected, we can show that susp A is (k+1)-connected and then show that loop susp A is k-connected. This gives us a k-connected map A -> 1 and a k-connected map loop susp A -> 1. This gives us a diagram which commutes with A and loop susp A in the top corners and 1 in the bottom. The LHS composition is homotopic to the RHS composition hence naming eta : A -> loop susp A, we have eta o unitmap being k-connected hence eta must also be k-connected. This isn't quite there.</div><div><br></div><div>Now I was hoping to use the fact that loop spaces are pullbacks hence there are maps coming out of 1s hence (k+1)-connectedness appears, but I couldn't get it to work.<br><br>On Sunday, 4 August 2019 12:59:09 UTC+3, Ali Caglayan wrote:<blockquote class="gmail_quote" style="margin: 0;margin-left: 0.8ex;border-left: 1px #ccc solid;padding-left: 1ex;"><div dir="ltr">Can we prove that the map A -> loop susp A is (k+1)-connected for a k-connected A, without having to invoke Freudenthal?<span style="font-size:15.7742px;font-family:sans-serif"></span></div></blockquote></div></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/a4549eab-f563-42ad-a95a-10166e8e0664%40googlegroups.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/a4549eab-f563-42ad-a95a-10166e8e0664%40googlegroups.com</a>.<br />